Frequency-drift compensation in chirped-pulse-based distributed acoustic sensing

ABSTRACT

Aspects of the present disclosure directed to frequency drift compensation for coded-DAS systems that use chirped pulses as a probe signal. Our inventive approach estimates timing jitter by correlating the amplitude of the estimated Rayleigh impulse response of every frame with a reference frame, and then re-aligns each frame by the estimated timing jitter. As the amount of timing jitter varies within a frame, every frame is divided into blocks where all samples have similar timing jitter, and perform timing jitter estimation and compensation on a block-by-block, frame-by-frame basis using an overlap-and-save method. Tracking of a slowly changing channel is enabled by allowing the reference frame to be periodically updated.

CROSS REFERENCE

This application claims the benefit of U.S. Provisional Pat. ApplicationSerial No. 63/270,199 filed 21OCT. 2021, the entire contents of whichbeing incorporated by reference as if set forth at length herein.

TECHNICAL FIELD

This disclosure relates generally to distributed fiber optic sensing(DFOS) systems, methods, and structures and, in particular, tofrequency-drift compensation in chirped-pulse-based distributed acousticsensing (DAS).

BACKGROUND

Recently, DFOS systems and methods have been employed to providesuperior acoustic and / or vibrational monitoring of roadways, bridges,and buildings. The reliability, robustness, and sensitivity of suchsystems is generally known to be unmatched by existing, legacy systemsand methods. Given such characteristics, further improvement of DFOS /DAS techniques - coupled with novel analysis systems and methods wouldrepresent a welcome addition to the art.

SUMMARY

An advance in the art is made according to aspects of the presentdisclosure directed to frequency drift compensation for coded-DASsystems that use chirped pulses as a probe signal.

Frequency drift results in timing jitter in the estimated Rayleighimpulse response, which is obtained by correlating the received Rayleighbackscatter with the originating chirp. Since in any given receivedframe, neighboring samples will have similar timing jitter, this effectcan be compensated by dividing each frame into small blocks where allsamples are shifted by the same timing jitter, correlating the amplitudeprofile from frame to frame, and then re-aligning them in time.

Our inventive approach provides an architecture for estimating andcorrecting timing jitter using an overlap-and-save architecture with lowalgorithmic complexity and enables coded-DAS systems to employ using“cheaper” lasers with larger frequency drift by using DSP to compensatefor the hardware.

Viewed from one aspect, our inventive approach estimates timing jitterby correlating the amplitude of the estimated Rayleigh impulse responseof every frame with a reference frame, and then re-aligns each frame bythe estimated timing jitter. As the amount of timing jitter varieswithin a frame, every frame is divided into blocks where all sampleshave similar timing jitter, and perform timing jitter estimation andcompensation on a block-by-block, frame-by-frame basis using anoverlap-and-save method. Tracking of a slowly changing channel isenabled by allowing the reference frame to be periodically updated.

Viewed from a first aspect, the present disclosure describes DFOSsystems, methods, and structures for monitoring outdoor cabinetscontaining fiber optic facilities in which the cabinet / fiber opticcable contained therein are configured to provide superior acousticsensing.

Viewed from a second aspect, the present disclosure describes DFOSsystems, methods, and structures for monitoring manhole structures.

Finally, viewed from yet another aspect, the present disclosuredescribes DFOS systems, methods, and structures employing a machinelearning-based analysis method that employs a temporal relation network.

BRIEF DESCRIPTION OF THE DRAWING

A more complete understanding of the present disclosure may be realizedby reference to the accompanying drawing in which:

FIG. 1(A) is a schematic diagram illustrating a DFOS system according toaspects of the present disclosure;

FIG. 1(B) is a schematic diagram illustrating a coded constant-amplitudeDFOS system with out-of-band signal generation according to aspects ofthe present disclosure;

FIG. 2(A) and FIG. 2(B) are a pair of plots illustrating the impact offrequency shift on the correlation of a chirped pulse for: FIG. 2(A) nofrequency shift, and FIG. 2(B) 4 MHz frequency shift according toaspects of the present disclosure;

FIG. 3 is a schematic diagram illustrating a canonical model of a coded-DAS interrogator where the same laser is used to generate probe signalx(t) and to serve as local oscillator for coherent detection of theRayleigh backscatter y(t) of the fiber under test (FUT) with Rayleighimpulse response h(t) according to aspects of the present disclosure;

FIG. 4 is a schematic diagram showing illustrative architecture forlaser frequency drift compensation in coded-DAS according to aspects ofthe present disclosure; and

FIG. 5 is a schematic diagram showing illustrative procedure forcompensating the time shift of each block according to aspects of thepresent disclosure;

DESCRIPTION

The following merely illustrates the principles of the disclosure. Itwill thus be appreciated that those skilled in the art will be able todevise various arrangements which, although not explicitly described orshown herein, embody the principles of the disclosure and are includedwithin its spirit and scope.

Furthermore, all examples and conditional language recited herein areintended to be only for pedagogical purposes to aid the reader inunderstanding the principles of the disclosure and the conceptscontributed by the inventor(s) to furthering the art and are to beconstrued as being without limitation to such specifically recitedexamples and conditions.

Moreover, all statements herein reciting principles, aspects, andembodiments of the disclosure, as well as specific examples thereof, areintended to encompass both structural and functional equivalentsthereof. Additionally, it is intended that such equivalents include bothcurrently known equivalents as well as equivalents developed in thefuture, i.e., any elements developed that perform the same function,regardless of structure.

Thus, for example, it will be appreciated by those skilled in the artthat any block diagrams herein represent conceptual views ofillustrative circuitry embodying the principles of the disclosure.

Unless otherwise explicitly specified herein, the FIGs comprising thedrawing are not drawn to scale.

By way of some additional background, we begin by noting thatdistributed fiber optic sensing (DFOS) is an important and widely usedtechnology to detect environmental conditions (such as temperature,vibration, acoustic excitation vibration, stretch level etc.) anywherealong an optical fiber cable that in turn is connected to aninterrogator. As is known, contemporary interrogators are systems thatgenerate an input signal to the fiber and detects / analyzes thereflected/scattered and subsequently received signal(s). The signals areanalyzed, and an output is generated which is indicative of theenvironmental conditions encountered along the length of the fiber. Thesignal(s) so received may result from reflections in the fiber, such asRaman backscattering, Rayleigh backscattering, and Brillionbackscattering. DFOS can also employ a signal of forward direction thatuses speed differences of multiple modes. Without losing generality, thefollowing description assumes reflected signal though the sameapproaches can be applied to forwarded signal as well.

FIG. 1(A) is a schematic diagram of a generalized, prior-art DFOS system. As will be appreciated, a contemporary DFOS system includes aninterrogator that periodically generates optical pulses (or any codedsignal) and injects them into an optical fiber. The injected opticalpulse signal is conveyed along the optical fiber.

At locations along the length of the fiber, a small portion of signal isreflected and conveyed back to the interrogator. The reflected signalcarries information the interrogator uses to detect, such as a powerlevel change that indicates - for example -a mechanical vibration. Whilenot shown in detail, the interrogator may include a coded DFOS systemthat may employ a coherent receiver arrangement known in the art such asthat illustrated in FIG. 1(B).

The reflected signal is converted to electrical domain and processedinside the interrogator. Based on the pulse injection time and the timesignal is detected, the interrogator determines at which location alongthe fiber the signal is coming from, thus able to sense the activity ofeach location along the fiber.

Those skilled in the art will understand and appreciate that byimplementing a signal coding on the interrogation signal enables thesending of more optical power into the fiber which can advantageouslyimprove signal-to-noise ratio (SNR) of Rayleigh-scattering based system(e.g. distributed acoustic sensing or DAS) and Brillouin-scatteringbased system (e.g. Brillouin optical time domain reflectometry orBOTDR).

Operationally, we assume that the DFOS system will beRayleigh-scattering based system (e.g., distributed acoustic sensing orDAS) and Brillouin-scattering based system (e.g., Brillouin optical timedomain reflectometry or BOTDR) with a coding implementation. With suchcoding designs, these systems will be most likely be integrated withfiber communication systems due to their lower power operation and willalso be more affected by the optical amplifier response time.

In the arrangement illustratively shown in the block diagram, we assumethat the coded interrogation sequence is generated digitally andmodulated onto the sensing laser via digital-to-analog-conversion (DAC)and an optical modulator. The modulated interrogation sequence may beamplified to optimal operation power before being directed into thefiber for interrogation.

Advantageously, the DFOS operation may also be integrated together withcommunication channels via WDM in the same fiber. Inside the sensingfiber, the interrogation sequence and the returned sensing signal may beoptically amplified - either via discrete (EDFA/SOA) or distributed(Raman) methods. A returned sensing signal is routed to a coherentreceiver after amplification and optical band-pass filtering. Thecoherent receiver detects the optical fields in both polarizations ofthe signal, downconverting them to 4 baseband lanes foranalog-to-digital conversion (ADC) sampling and digital signal processor(DSP) processing. As those skilled in the art will readily understandand appreciate, the decoding operation is done in the DSP to generatethe interrogated Rayleigh or Brillouin response of the fiber, and anychanges in the response are then identified and interpreted for sensorreadouts.

With continued reference to the figure, since the coded interrogationsequence is generated digitally, the out-of-band signal is alsogenerated digitally, and then combined with the code sequence beforewaveforms are created by the DAC. When generated together digitally, theout-of-band signal will only be generated outside the time period of thecode sequence, so when added together, the combined waveform will have aconstant amplitude.

In distributed acoustic sensing (DAS), the interrogator launches a probesignal x(t) into the fiber under test (FUT) to estimate its Rayleighimpulse response h(t) . The received signal is given by the convolutiony(t) = x(t)⊗h(t) . Conventionally,

$x(t) = \sqrt{P}\,\text{rect}\left( {t/T} \right)$

is a pulse with peak power P and duration T.

In order to measure time-variation in the Rayleigh impulse responsecaused by acoustic vibration, the interrogator transmits x(t)periodically at a frame rate of T_(p). Provided T_(p) is longer than theround-trip propagation time T_(rt) of the FUT, the received signal is asequence of optical time-domain reflectometry (OTDR) traces.

The spatial resolution of y(t) is z_(res) = (c/2n_(eƒƒ))T and iscontrolled by the bandwidth of x(t). By measuring the phase relationshipbetween pairs of points at positions z₁ = (c/2n_(eƒƒ))t₁ and z₂ =(c/2n_(eƒƒ))t₂ in the received OTDR, and how that phase varies withtime, the time-varying longitudinal strain between z₁ and z₂ can bemonitored. This technique of phase-OTDR (ɸ-OTDR) is well known in theart.

Due to the weak power of Rayleigh backscatter, the reach achievable byDAS is limited. Coded-DAS increases the signal-to-noise (SNR) ratioachieved by DAS to allow for longer FUT. Instead of launching pulses -and as previously noted - the probe signal in coded-DAS is a sequencewith autocorrelation function r_(xx)(t) = x(t) ∗ x(t) that is as closeto a delta function as possible, with its width constrained only by thebandwidth of x(t). The coded-DAS interrogator performs correlation ofthe received signal with x(t) to obtain z(t) = x(t) ∗ y(t) = rxx(t)⊗h(t). Since rxx(0) =

∫_(−∞)^(+∞)|x(t)|²

dt = P ▪ T_(c) is the energy of x(t), the longer the sequence, thelarger the received signal. Advantageously, SNR can be increasedlinearly with T_(c) without sacrificing spatial resolution. Laser phasenoise is the limiting factor for how large T_(c) can be.

One well-known family of sequences with good autocorrelation propertiesare chirped pulses:

$x(t) = \sqrt{P}\exp\left( {j2\pi\alpha\frac{t^{2}}{2}} \right)\text{rect}\left( \frac{t}{T_{c}} \right)$

where T_(c) is the chirp duration, α is the chirp rate, and

$\sqrt{P}$

is the amplitude of the envelope. The bandwidth of the x(t) is B =αT_(c), and its autocorrelation is given by:

R_(xx)(t) = x(t) * x(t)= (T_(c) − |t|) sinc(αt(T_(c) − |t|)),

For long chirp duration T_(c) » T, the width of the main lobe of thesinc in Eq. (2) is T = 1/αT_(c) = 1/B, which is the same spatialresolution as a rectangular pulse of the same bandwidth used inconventional OTDR.

Chirped pulses have the special property that the correlation functionbetween two chirped pulses x₁(t) and x₂(t) is only the sinc function inEq. (2) if their chirp rates α₁ and α₂ match. If this is the case, theircorrelation peak occurs where their center frequency match.

FIG. 2(A) and FIG. 2(B) are a pair of plots illustrating the impact offrequency shift on the correlation of a chirped pulse for: FIG. 2(A) nofrequency shift, and FIG. 2(B) 4 MHz frequency shift according toaspects of the present disclosure. Chirps have the special property thata correlation peak is produced when their instantaneous frequenciesalign. Frequency modulation of one chirped pulse thus results in atemporal shift of the correlation peak (and also a slight broadening ofthe main lobe of the correlation function due to reduced bandwidthoverlap).

An example is shown in FIG. 2(A) and FIG. 2(B) where a chirped pulse ofduration T_(c) = 50 µs and bandwidth B = 10 MHz (α = 2 × 10¹¹ s⁻²) iscorrelated with the same chirped pulse that is frequency-shifted by Δv =4 MHz. It is observed that their correlation x₂(t) ∗ x₁(t) is centeredat an offset of Δτ = Δν/α = 20 µs . In addition, the width of thecorrelation is inversely proportional to their bandwidth overlap, whichin the example shown is 6 MHz.

This property of chirped pulses, where frequency modulation results intemporal shift of the center of the correlation function is important tocoded-DAS based on chirped pulses.

FIG. 3 is a schematic diagram illustrating a canonical model of a coded-DAS interrogator where the same laser is used to generate probe signalx(t) and to serve as local oscillator for coherent detection of theRayleigh backscatter y(t) of the fiber under test (FUT) with Rayleighimpulse response h(t) according to aspects of the present disclosure.

Consider the canonical model of a DAS system shown in FIG. 3 , where thesame laser is used to generate the probe signal and to serve as localoscillator (LO) for coherent detection of the Rayleigh backscatter.Consider the reflection from the point shown in the FUT in FIG. 3 .Since there is a propagation delay difference of τ_(z) = (2n_(eff)/c)zbetween the reflected signal and the LO, the laser frequency will havedrifted over duration τ_(z). Frequency drift is the manifestation oflaser phase noise in the low frequency region and is caused by factorssuch as temperature fluctuation and mechanical vibration, etc., whichcauses fluctuation in the optical length of the laser cavity, thusimpacting the laser’s center frequency.

As per FIG. 2(A), and FIG. 2(B), the impact of frequency drift isuncertainty in time (or position) of the Rayleigh impulse response.Normally, the fiber position z is mapped to time coordinate t =(2n_(eff)/c)z in the Rayleigh impulse response h(t). But due to laserfrequency drift, the actual fiber position at time t is z̅ + δz where z̅ =(c/2n_(eyy))t is the mean position, and δZ is a variable that changesfrom frame to frame, depending on the laser frequency drift Δv duringthat frame. When DAS computes a <p-OTDR based on the differentialproduct between two points separated by gauge length z_(g), instead ofmonitoring the phase difference between two fixed scatterers at z₁ = z̅ –z_(g)/2 and z₂ = z̅ + z_(g/)2, the actual phase difference is computedfor two scatterers at z₁ = z̅ + δz – z_(g)/2 and z₂ = z̅ + δz + z_(g)/2.If δZ is larger than the spatial resolution, the resulting DAS willbecome non-sensical, since scatters separated by larger than the spatialresolution should be independent.

The impact of laser frequency drift on the performance of DAS isunderstood in an experiment in which a 50-km long FUT was probed usingchirped pulses of duration T_(c) = 50 µs and bandwidth B = 10 MHz(spatial resolution z_(res) = c/2n_(eff) B ≈ 10 m), at a repetition rateof 1 kHz. At 50 km, a piezo-electric transducer (PZT) was inserted,followed by another 100 m of termination fiber. The PZT was excited witha 67-Hz sine wave with peak-to-peak amplitude of 1.6 rad.

When one employs a specific (“bad”) laser, the frequency drift resultsare so severe that the background noise level is larger than thevibration amplitude of the PZT. With a different (“good”) laser however,the 67-Hz sine wave is easily observed.

Although it is possible to build a chirped pulse DAS using the “good”laser, it is more expensive. Since frequency drift only results inlinear translation of the time/position axis, and fiber positions closetogether should experience similar time/position shift as the laserfrequency should be stable over short timescales, it should be possibleto compensate this time (position) jitter using digital signalprocessing (DSP), and therefore allow DAS using the “bad” laser withhigher phase noise.

Notwithstanding, the present disclosure provides a solution forfrequency drift compensation for coded-DAS systems that use chirpedpulses as the probe signal. Frequency drift results in timing jitter inthe estimated Rayleigh impulse response, which is obtained bycorrelating the received Rayleigh backscatter with the originatingchirp.

Since in any given received frame, neighboring samples will exhibitsimilar timing jitter, our inventive disclosure compensates for thiseffect by dividing each frame into small blocks where all samples areshifted by the same timing jitter, correlating the amplitude profilefrom frame to frame, and then re-aligning them in time. Our disclosedsolution provides an architecture for estimating and correcting timingjitter using an overlap-and-save architecture, and also providespreferred implementations with low algorithmic complexity. Our disclosedsolution enables coded-DAS using “cheaper” lasers with larger frequencydrift by using DSP to compensate the hardware deficiencies.

As will become apparent to those skilled in the art, features of ourdisclosed solution according to aspects of the present disclosureinclude: (i) estimation of timing jitter by correlating the amplitude ofthe estimated Rayleigh impulse response of every frame with a referenceframe, and then (ii) re-aligning each frame by the estimated timingjitter.

As the amount of timing jitter varies within a frame, our inventiveapproach divides every frame into blocks where all samples exhibit asimilar timing jitter, and then perform timing jitter estimation andcompensation on a block-by-block, frame-by-frame basis using anoverlap-and-save method. Our inventive approach also enables tracking ofa slowly changing channel by allowing the reference frame to beperiodically updated.

FIG. 4 is a schematic diagram showing illustrative architecture forlaser frequency drift compensation in coded-DAS according to aspects ofthe present disclosure. As illustrated, the architecture shown in FIG. 4is for chirped pulses using an overlap-and-save algorithm.Operationally, after correlating the received signal with the knownchirp to obtain h_(j)(t) = x(t) ∗ y_(j)(t) in the j-th frame, the timeaxis is partitioned into N_(b) overlapping blocks of duration T_(b) andoverlap T_(ov). Taking frame j = 0 as reference, the square amplitude ofeach block |h_(j)(t_(b))|² in subsequent frames is compared with|h₀(t_(b))|² to find their relative offset τ_(j),_(b). Each block isthen shifted by τ_(j,b) to obtain the compensated outputs h_(j)(t_(b)).Normal DAS operations are then performed on h̃_(j)(t).

To further clarify, we let y_(j)(t) be the complex-valued signal vectorreceived due to the probe pulse launched in frame j. Let h_(j)(t) = x(t)∗ y_(j)(t) be the output of the correlator. The duration of each vectoris T_(rt) equal to the round-trip return time of the FUT. We divideh_(j)(t) into overlapping blocks of duration T_(B) and overlap T_(ov).The number of overlapping blocks N_(b) needs to satisfy (N_(b) –1)(T_(b) – T_(ov)) + T_(b) ≥ T_(rt). Let t_(b) ε {t: b(Tb – T_(ov)) ≤ t< b(Tb – T_(ov)) + Tb} be the duration of block b, where 0 ≤ b < N_(b),

We then take the initial frame h_(o)(t) to be the reference. For everyblock b, we correlate the amplitude |h_(j)(t_(b))|² of all subsequentframes with the reference |h₀(_(tb))|² to estimate the timing jitterτ_(j,b) = max ∫_(t) _(b) |h_(j)(t ^(_) _(τ))|² |h₀(t)|² dt during blockb of frame j. We then shift h_(j)(t_(b)) by τ_(j,b) to obtain thecompensated signal h_(j)(t_(b)) = h_(j)(t_(b) – τ_(j,b)). The usefulpart of the overlap-and-save, b(Tb – T_(ov)) ≤ t < (b + 1)(T_(b) –T_(ov)), is then stored. The frequency-drift compensated signal is h̃(t).

The downstream operations needed to estimate vibration at every point inthe FUT are then the same as conventional DAS. These operations mayinclude calculating differential beat products at a pre-defined gaugelength, diversity-combining the beat products from differentpolarizations, frequencies, and spatial channels, etc., and then finallytaking the unwrapped phase at every location in the FUT.

According to our inventive disclosure, the block length T_(b) should bechosen so that the root mean square (r.m.s.) frequency drift(σ_(Δv)(T_(b)) over duration T_(b) is much less than time resolution ofthe chirp T = 1/B, i.e., the samples in h_(j)(t_(b)) are all delayedrelative to h_(o)(t) by a similar amount down to an accuracy of thetime-resolution 1/B. T_(b) will depend on the laser used by theinterrogator.

Suppose the laser’s instantaneous frequency is v(t). Let S_(vv)(f ) bethe two-sided frequency noise spectrum of is v(t). The frequency driftΔv_(T) _(b)(t) = v(t) - v(t – T_(b)) between two time instancesseparated by T_(b) has the Fourier transform

Δv_(T_(b))(f) = [1 − e^(−j2πfT_(b))]v(f) .

. Hence, the spectrum of Δv_(T) _(b)(t) is

S_(Δv_(T_(b)))_(Δv_(T_(b)))(f) = |1 − e^(−j2πfT_(b))|²S_(vv)(f).

It is well known that the variance

E[|Δv_(T_(b))(t)|²]

is equal to the autocorrelation

R_(Δv_(T_(b))Δv_(T_(b)))(τ)

evaluated at τ = 0, which is equal to the integral of

S_(Δv_(T_(b)))_(Δv_(T_(b)))(f)

over all frequencies. Hence,

E[|Δv_(T_(b))(t)|²] = (∫_(−∞)^(+∞)|1−

(e^(−j2πfT_(b))|²S_(vv)(f)  df.

Since Δv_(T) _(b)(t) is a Gaussian process, T_(b) should be chosen sothat

$\sigma_{\Delta v}\left( T_{b} \right) = \sqrt{E\left\lbrack \left| \Delta_{V_{T_{b}}} \right|^{2} \right\rbrack} = {1/{\varsigma B,}}$

to ensure the probability

$\text{erfc}\left( {\varsigma/\sqrt{2}} \right)$

is small that the samples T_(b) at the beginning and end of a block haverelative timing jitter greater than 1/B.

Similarly, the overlap duration T_(ov) should be larger than twice themaximum value of |τ_(j,b)|that may need to be compensated. As per FIG. 3, the r.m.s. value of the timing jitter is σ_(Δv)(t)/a, which increaseswith distance from the interrogator z = (c/2n_(eyy))t,. If a constantoverlap size is used, then the overlap duration should be chosen as

$T_{ov} = K\frac{\sigma_{\Delta v}\left( T_{rt} \right)}{\alpha},$

to ensure the probability

$\text{erfc}\left( {\kappa/\sqrt{2}} \right)$

is small that the time jitter τ_(j,N) _(n) of the last block of anyframe is larger than the overlap duration.

In practice, the Rayleigh impulse response h(t) that is estimated byh_(j)(t) is slowly changing with time due to polarization rotation,temperature fluctuation, etc. If the reference h₀(t) is held for toolong, then eventually, h(t) will have changed so much that |h₀(t)|² isno longer a good match for the incoming frames. Hence, the referenceshould be updated regularly every N_(u) frames. i.e., the mN_(u)-thtiming-jitter-compensated frame h̃_(mn) _(u)(t) should be used as the newreference for compensating frames h_(mN) _(u+1)(t) to h(m₊₁)_(N)_(u)(t).

FIG. 5 is a schematic diagram showing illustrative procedure forcompensating the time shift of each block according to aspects of thepresent disclosure. Correlation with the reference is first computed tofind the index n_(j),b at which r_(jo),_(b)[m] is maximized over –N_(ov)/2 ≤ m ≤ N_(ov)/2 . The block is then barrel shifted by n_(j),_(b)samples.

If more accurate compensation is required, the ratios between the twoneighboring samples with the peak sample can be computed, and a lookuptable can be used to find the fractional shift n'_(j,b). Fractionalshifting of the block is achieved using an interpolation filter.

Reduced Complexity Implementation

Per the aforementioned description, the computationally expensiveoperations are: (a) correlating each block |h_(j)(t_(b))|² with|h₀(t_(b))|², and (b) shifting each h_(j)(t_(b)) by τ_(j,b). Inpractice, these signals are sampled at a rate of T_(S) = 1/MB where M isthe oversampling ratio relative to the chirp bandwidth B. Similar todigital coherent receivers for telecommunications, M = 2 is often chosenas compromise. Note that τ_(j,b) does not have to be an integer multipleof T_(s).

The digital correlation

r_(j0, b)[m] = ∑_(n ∈ n_(b))|h_(j)[n − m]|²|h₀[n]|²

is expected to be a sinc²(^(.)) function sampled at T_(s). Acomputationally simple solution is to compute r_(jo),_(b)[m] only for arange of indices – N_(ov)/2 ≤ m ≤ N_(ov)/2, where N_(ov) = [T_(ov)/T] isthe overlap duration in number of samples, and then find the indexn_(j),b which maximizes r_(jo,b)[m] over this range. The signals h_(j)[n_(b)] are then barrel shifted by n_(j),b samples, which does notrequire any multiplications. This simplified implementation aligns allh_(j)[n_(b)] to the nearest sample, ensuring the compensated signalshave residual timing jitter of at most ±0.5T_(S).

Note that it is possible to reduce timing jitter further with analternative implementation having slightly higher computationalcomplexity. Since

r_(j0, b)[m]=

∑_(n ∈ n_(b))|h_(j)[n − m]|²|h₀[n]|²

is a sampled sinc²(^(.)) function with the width between first nulls ±Msamples apart, we can take the index n_(j),b which maximizesr_(jo),_(b)[m], along with its two neighbors n_(j),b – 1 and n_(j),b +1.

A lookup table indexed by the two ratios r_(j0,b)[n_(j),_(b) –1]/r_(j,0),_(b)[n_(j,b)] and r_(jo),_(b)[n_(j,b) + 1]/r_(j0,b)[n_(j,b])quantized to a given timing jitter accuracy can then be used to find thefractional sample offset n'_(j,b') where

$n_{j,b} + {n^{\prime}}_{j,b} = \frac{\tau_{j,b}}{T_{s}}.$

As with the previous case, h_(j)[n_(b)] is first barrel-shifted byn_(j),b samples. To shift the remaining fractional sample n'_(j,b') aninterpolation filter h_(int)[n] of N_(int) taps can be used. Thecoefficients for h_(int)[n] can once again be stored in a lookup tablefor different fractional offsets, quantized to a timing jitter accuracyrequired. This step is analogous to timing recovery in digital coherentreceivers for telecommunications, where the received signal is digitallyresampled using interpolation filters to keep the samples synchronizedwith the symbol clock. As N_(int) can be small (e.g., 3 or 5 taps),resampling using interpolation filters can be of low complexity equal toN_(int) complex multiplications per sample per polarization.

Example Experimental Results

We have produced experimental results demonstrating the operation of thefrequency-drift compensation according to the present disclosure. Theexperimental setup is as shown in FIGS. 2A 2B, where chirped pulses ofduration T_(c) = 50 µs and bandwidth B = 10 MHz (spatial resolutionz_(res) = c/2n_(eff) B ≈ 10 m), at a repetition rate of 1 kHz aregenerated using a digital-to-analog converter (DAC) which drives aMach-Zehnder I/Q modulator.

The FUT comprises of a 50 km spool of standard single-mode fiber (SSMF),followed by a 12-m long piezo-electric transducer (PZT), whose output isterminated by a 100-m long fiber. The PZT is driven with a 67-Hz sinewave with peak-to-peak amplitude of 1.6 rad. It was observed thatdistortion caused by frequency drift is so severe that the resultingbackground noise level is larger than the vibration amplitude of the PZT

To evaluate timing jitter associated with laser frequency drift, weevaluated over 200 traces of |h_(j)(t) |² recovered after correlatingthe received signal with the chirp. An exponential decay profile due tofiber attenuation, which has a round-trip loss of ≈ 0.4 dB/km wasproduced. As expected, there is negligible timing jitter between thebeginning of the fiber, but timing jitter of around 1.5 µs peak-to-peakis observed at the end of the FUT, which corresponds to a frequencydrift of ~ 300 kHz peak-to-peak.

With this jitter characterized, we used our inventive method tocompensate frequency drift. Block sizes of T_(b) = 25 µs and overlapduration of T_(ov) = 2.5 µs were used. When we applied this technique todata as above, |h_(j)(t)|² near the end of the after timing-jittercompensation, the 200 traces become well aligned with each other.

When our technique is applied to performance of DAS using the“cheap/bad” laser previously described with frequency driftcompensation, the 67-Hz vibration is now clearly observed and the phasespectrum has a noise floor corresponding to a strain sensitivity of 38

$\text{p} \in \text{/}\sqrt{\text{Hz}}.$

Finally, the performance of DAS using a “good” laser that exhibits muchless frequency drift than the “bad” laser produces a strain sensitivityat a position of the PZT also around 38

$\text{p} \in \text{/}\sqrt{Hz}.$

Thus, frequency-drift compensation enables the cheaper “bad” laser withlarger frequency drift to achieve comparable DAS performance as theexpensive “good” laser

At this point, while we have presented this disclosure using somespecific examples, those skilled in the art will recognize that ourteachings are not so limited. Accordingly, this disclosure should onlybe limited by the scope of the claims attached hereto.

1. A method for frequency-drift compensation in a chirp-pulse-based distributed acoustic sensing system (DAS) comprising: a length of optical fiber sensor cable; a DAS interrogator in optical communication with the length of optical fiber sensor cable, the DAS interrogator including: a seed laser; and a coherent receiver; the interrogator configured to: produce probe signals including chirped pulses at a given frame rate, chirp duration and chirp slew rate and launching the probe signals into the length of optical fiber sensor cable; and recover Rayleigh backscatter from the optical fiber sensor cable using the coherent receiver; wherein the seed laser exhibits a non-negligible frequency drift such that a frequency of a local oscillator is different from a frequency which generated the chirped pulses resulting in timing jitter of a Rayleigh impulse response determined from a correlation of received backscatter with known chirp; the method comprising: compensating timing jitter in an estimated Rayleigh impulse response by dividing each received frame of Rayleigh backscatter into overlapping blocks wherein the first frame received is a reference frame from which timing jitter of subsequent frames are estimated by correlating amplitude profiles of the subsequent frames with an amplitude profile of the reference frame and; re-aligning the frames by the estimated jitter.
 2. The method of claim 1 wherein a timing jitter of the reference frame is periodically updated to track a slowly changing channel.
 3. The method of claim 2 wherein the timing jitter is estimated to an accuracy equal to an integer multiple of signal samples such that a time re-alignment corresponds to a barrel shift of the samples in a register.
 4. The method of claim 3 wherein the timing jitter is estimated to an accuracy equal to a fraction of a signal sample.
 5. The method of claim 4 wherein the fraction of the signal sample is obtained using a lookup table indexed by a ratio between a peak correlation value and correlation values of its neighbor peaks.
 6. The method of claim 5 wherein resampling in time is performed by convolving each block of each frame of the Rayleigh impulse response by interpolation filter. 